I occasionally have this issue with my iphone 6 (and my iphone 5 before that). If you swipe up from the bottom to get into control center, on the lower right is where the audio is going. It will say "Volt" but nothing is playing. Press that and switch it to "iphone" and the sound will start coming back out of the phone's speaker. Then switch it back to "Volt" and it's worked 100% of the time for me. Takes just a few seconds. May be faster than your current method? Worth a try...
OP... bluetooth is supposed to be standard, so if both devices claim to be bluetooth, they are supposed to work together. For some reason, car and phone makers (not sure who is more to blame) sometimes implement the standard in a non-standard way and it can cause issues. See my comment above, perhaps your Pixel is doing something similar, can you switch between Volt and phone speaker output and back and see if that fixes it? Did you make sure the volume is not at 0 on the phone? Have you verified the pixel works with other bluetooth devices (speakers, headsets, etc)?